Optimal stock allocation in single echelon inventory systems subject to a service constraint

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1 Abstract number: Optimal stock allocation in single echelon inventory systems subject to a service constraint Annalisa Cesaro Dario Pacciarelli Dipartimento di Informatica e Automazione, Università Roma Tre, Via della Vasca Navale, 79, Rome I-00146, Italy cesaro@dia.uniroma3.it pacciarelli@dia.uniroma3.it phone: POMS 20th Annual Conference Orlando, Florida U.S.A. May 1 to May 4, 2009 This paper deals with spare optimization in multi-location inventory systems with single item and repairable spare parts. Lateral and emergency shipments occur in response to stockouts. A continuous review basestock policy is assumed for the inventory control of the spare parts. The resulting model is a queueing network with blocking, which can be studied using a Markov chain modeling approach. The objective is to minimize the total costs for inventory holding, lateral transshipments and emergency shipments subject to a target level of operational availability of the whole system. We model the stock allocation problem as a non linear integer program and exploit its special structure to design a primal-based and dual-based algorithm. Our solution procedure is based on a convergent Lagrangian and objective level cut method. Computational experiments, carried on practical data from an airport equipment maintenance context show that this method finds an optimal allocation for all practical instances for which the optimum is known. 1. Introduction Single echelon inventory systems are experiencing an increasing interest in practice, in particular for the management of expensive spare parts. In such a context, the supply chain involves at least three actors: equipment users, logistics companies and equipment suppliers. The users need spare parts to carry on their business without interruptions. Intermediate logistic companies are in charge of replenishing spare parts in the short term, by granting 1

2 the contractual service level to the users at minimum cost. The suppliers are responsible for supplying new components and/or repaired items to the logistic companies. As observed by several authors, see e.g. by [10], the logistics of spare parts differs from those of other materials in several ways. Equipments may have remarkable costs, long repairing times and sporadic failures. The latter are difficult to forecast and may cause relevant financial effects, due to the economical implications of a lack of equipment at the operational sites. In such cases, continuous review policies are typically adopted to reduce both reaction time to stockouts and inventory levels [2, 9]. Several heuristic procedures can be found in the literature for allocating spares to warehouses in a single echelon context with complete pooling [3, 12, 21]. This paper proposes and evaluates a new solution procedure for spare parts allocation in this context. Our work is motivated by a practical problem faced by a large Italian logistics company. The company handles 17 warehouses supporting the daily activity of 38 airports spread over the Italian territory. Stock levels are currently determined with the VARIMETRIC algorithm of Sherbrooke (Sherbrooke 2004), based on a stiff hierarchic structure. However, in operation lateral transshipments take place between stocking points whenever there is an emergency request for a spare part, using couriers and overnight carriers to rapidly transfer the required items. The company is therefore interested in determining the potential savings deriving from explicit inclusion of lateral transshipments in the model. To this aim, a new method for stock level definition and spare parts allocation is necessary, and this need motivates the present paper. Specifically, this paper develops a new method for finding the optimal stock allocation in a single echelon inventory system with continuous review, lateral and emergency transshipments, non-negligible transshipment times, subject to a minimum level of operational availability (e.g. the fraction of time during which all operational sites are working). Our optimization method is independent from the specific method chosen for assessing the operational availability of a given allocation, and therefore it applies as well to different contexts with different performance indicators. This paper is organized as follows. Relevant notation is described in Section 2. Section 3 describes the single echelon one-for-one ordering model with complete pooling. In Section 4 the spare parts allocation problem is formulated as a non-linear integer program, and a convergent Lagrangian and objective level cut method is described for finding an optimal allocation. Computational experiments are presented in Section 5, based on practical data from the airport maintenance context. Some conclusions follow in Section 6. 2

3 2. Notation In order to formally define the problem, let us introduce the following notation. Let A = {1, 2,..., a} be a set of operational sites (e.g., airports) where working equipments are located. We assume that operational sites are grouped on a regional basis, with a warehouse of spare parts for each regions. Let W = {1, 2,..., w} be the set of regional warehouses. Let s i be the number of spare parts to allocate to each warehouse i W, and with s = (s 1,..., s w ) an allocation of spares to warehouses, i.e., the vector of decision variables. Let µ be the service rate of a server at warehouse h, T hi be the transfer time for a spare from warehouse h to warehouse i and T s (j, h) be the substitution time, i.e., the time needed to transfer a spare part to the site j A from the warehouse h W and to physically replace the failed item. Let λ jh be the rate of requests for a spare part from site j to warehouse h, λ h = j A λ jh be the aggregated arrival rate of failure processes at warehouse h and λ h the effective arrival rate at warehouse h including lateral transshipments. Let P B (s) be the network blocking probability for a given allocation s, i.e., the probability that a failure occurs at some site and no warehouse can satisfy the spare demand. Given an allocation s, let π hi (s) be the probability of the event: there are no spares in warehouse h W and i W is the closest warehouse with available spares (i.e., every warehouse l such that T hl < T hi, including the case l = h, is in stockout condition). Let n = (n 1,..., n w, n w+1 ) be a vector representing the state of the network, in which n i is the number of outstanding requests at warehouse i W, and n w+1 is the number of outstanding emergency requests to the external supplier. In our system, n w+1 > 0 only if n i = s i for all i W. Let p(n) be the probability of being in state n for the whole warehouses network. Denote p i (s i ) as the marginal probability of having s i outstanding orders at warehouse i, i.e., the probability of having no stock available at warehouse i. We also denote with MT T R the mean time to repair, i.e. the average replenishment time of the external supplier, with M T BF the mean time between failures, with OS the order and ship time, with MCMT the mean corrective maintenance time and with OA the operational availability of all the a sites. Let c h be the inventory holding cost for warehouse h, c tr ij be the cost for a lateral transshipment from warehouse j to warehouse i, in stockout condition, and c em be the emergency transshipment cost. 3

4 3. Problem description In our model, a logistic company wants to find the stock level s i of each warehouse i W such that a minimum level of service is provided at the operational sites and the overall cost is minimum. Costs are related to inventory holding, transshipments and emergency transshipments. Given an allocation s of spares to warehouses, the model we use to compute the level of service is a single item, single echelon, w-locations, continuous review, one-for-one replenishment policy inventory system, which allows for lateral and emergency transshipments with complete pooling and non-negligible transshipment times in operational availability computations. The resulting model is a queueing network with blocking, in which each warehouse i W is viewed as a queue with s i servers and no buffer and the external supplier is viewed as a queue with infinitely many servers. 3.1 Performance indicator The contractual service level to grant is the operational availability (OA) of all operational sites for each item, computed as in [17]. OA = MT BF MT BF + MCMT. (1) In equation (1), MCMT is the average time occurring from the failure of an item to its physical substitution. This is the substitution time if the spare is available at the regional warehouse. If no spares are locally available, the request is forwarded to the closest warehouse with available spares and MCMT increases by the deterministic transfer time between the two warehouses. When no warehouse has spares available, MCMT equals the substitution time plus the replenishment time from the external supplier. The MCMT can be therefore computed as follows: MCMT = h W j A λ jh T s (j, h)+ h W (λ h i W π hi(s) T ih )+ ( h W λ h) P B (s) (MT T R + OS) We observe that the first term h W A j=1 λ jh T s (j, h) of Equation (2) only depends on the failure process and on the distance between the sites and their respective regional warehouses. In other words, it does not depend on the specific spare parts management (2) 4

5 policy being used. Therefore, for sake of simplicity we assume it negligible in our model and omit its computation in the rest of this paper. As for the quantity π hi (s) defined in Section 2, we assume that a strict deterministic nearest chosen neighbor rule is adopted for sourcing a lateral transshipment, as in Kukreja [12]. Differently form [12], we use equation (3) to compute this value, which directly follows from the definition of π hi (s). π hi (s) = (1 p i (s i )) (p l (s l )). (3) l:t hl <T hi 3.2 Assumptions We use the Poisson distribution for the demand process, which is a typical assumption for modeling low demand processes [18]. The MTBF of an equipment depends on its age and on other exogenous agents, such as the damp, the temperature and other operational conditions. Therefore, we use specific values for each site. The replenishment time of the external supplier is a random variable, exponentially distributed, with known mean value MTTR. The capacity of the supplier repair shop is assumed to be infinite. It follows that also the number of replenishments from the external supplier follows the Poissonian distribution. These common assumptions make possible to use the Markovian analysis for modeling the multi-dimensional inventory system. However, as shown by [12], the results obtained with these assumptions holds under less restrictive hypothesis on the replenishment distribution. 3.3 Multi-dimensional markovian model In order to compute the operational availability of the system under study for a given allocation s, we must compute the probability p(n) of each state n = (n 1,..., n w, n w+1 ) of the associated Markov chain, defined in Section 2. To this aim we use a model similar to that of Wong et al. [19]. The main difference is that we include the external supplier explicitly in the model, while in [19] a failure of a part occurring when all warehouses are in stockout condition is lost. In case of blocked network, the first repaired item returned by the external supplier is used for replacing a failed item in some operative site, if any. There are direct transitions among states just in case of a single arrival event (i.e. a request for a spare at some warehouse) or a single departure event (i.e. the replenishment of a repaired item by the external supplier). 5

6 Figure 1: The infinite state space Markov chain for a system of two warehouses (a) and finite space state model (b) and the aggregated birth death model (c) Let e i be a vector with w + 1 elements, all equal to 0 but the element in position i that is equal to 1, and let ψ(h, i) be equal to 1 if (n i < s i and n l = s l for each l W such that T lh < T ih, included l = h) and be equal to 0 otherwise. With this notation, n+e i is the state of the Markov chain representing an arrival at the i-th warehouse (if i W and n i < s i ), due either to a failure in the i-th service region or to a re-forwarded request from some other warehouse h in stockout conditions, i.e., such that ψ(h, i) = 1. Similarly, n e i is the state with a departure from the i-th warehouse (if i W and n i > 0). For the external supplier, n + e w+1 represents a new emergency request (if n i = s i for each i W ) and n e w+1 represents the fulfillment of an emergency request (if n w+1 > 0). The transition rate q(n, m) from state n towards state m = n ± e i and n ± e w+1 is as follows. q(n, n + e i ) = λ i + h W {i} ψ(h, i) λ h, for i W and n i = 0, 1,..., s i 1; q(n, n + e w+1 ) = i W λ i, if n i = s i i W ; q(n, n e i ) i µ, for i W and n i > 0 and n w+1 = 0; q(n, n e w+1 ) = w+1 i=1 n i µ, for n i = s i i W and n w

7 Figure 1(a) shows the Markov chain for two warehouses, the first having two spares and the second having three available spares. Steady state probabilities can be computed for each state in the Markov chain by solving a linear system. Note that, to this aim the proposed Markov chain is equivalent to the one shown in figure 1(b), having a finite number of states. In the latter Markov chain, all states in which all warehouses are in stockout condition are grouped in one single state with different departure transition rates. Let n B be the state in the original model where i W, (n B ) i = s i, (n B ) w+1 = 0 and p(n) be equal to the probability of being in state n. Denote P B (s) as in Section 2. In the equivalent model the modified departure rates of the single state corresponding to n B and all other blocked states in the original model become: q(n B, n B e i ) = (n B ) i µ F for i W and (n B ) i = s i and (n B ) w+1 = 0 and by using F = p(n B) P B (s). p(n B) and P B (s) may be easily computed, as stated in Theorem 1 in [6],by using the equivalence with the proposed Markov chain model with a simple birth death model, as shown in figure 1(c). Specifically let S = i W s i W i be the total stock level in the network, and let ρ = λ i. Given a set W of µ warehouses, with total stock level S, in which the service process is exponentially distributed with average rate µ for each server and the demand flow to warehouse i W is Poissonian with average rate λ i, the blocking probability of all warehouses is P B (s) = 1 S 1 k=0 ρ k k! e ρ. Finally p(n B ) is equal to ρn B N B! e ρ, where N B = w+1 i=1 (n B) i. By using the state probabilities computed solving the above finite state Markov chain model, we are able to compute p i (s i ) for each i W. p i (s i ) is the marginal blocking probability for the i-th warehouse, as defined in Section 2. It may computed as follows: p i (s i ) = p(n) (4) n N:n i =s i where N is the Markov chain state space. 4. Solution procedure In this section we face the main problem of defining an optimal allocation of spares to warehouses. We model the spare part optimization problem as a non linear program and propose a primal-based, dual-based algorithm for solving it at optimality. To this aim we study a Lagrangian relaxation of our model and use it within an iterative procedure known as convergent Lagrangian and objective level cut method [15]. 7

8 4.1 Problem formulation The Spares Allocation Problem is the problem of finding an allocation s which minimizes the overall cost for inventory holding, lateral and emergency transshipments, subject to a constraint on the minimum operational availability of the system, defined by equation (2) with the assumption A h W j=1 λ jh T s (j, h) = 0. Letting L be the minimum operational availability level that must be satisfied by a feasible allocation, the Spares Allocation Problem P 0 can be formulated as an integer program with non-linear objective function and a single non-linear constraint. With the notation defined in Section 2 we have: Problem P 0 : min s.t. : w i=1 ch s i + λ i j W π ij(s) c tr ji + λ i P B (s) c em w i=1 [λ i j W π ij(s) T ji + λ i P B (s) (MT T R + OS)] > (1 L) MT BF L (5) In the following subsections we focus on the computation of an optimal allocation s. The solution method consists of an iterative procedure. At the k-th step, the Lagrangian relaxation of a revised problem P k is computed. Problem P k is obtained from the original problem P 0 by adding constraints on the objective value, such that the optimum still satisfies all the constraints and the difference between the Lagrangian lower bound of problem P k and the current best upper bound is reduced. This method is called convergent Lagrangian and objective level cut method [15]. Let us briefly recall the main concepts of the method. Given a set X R n and r+1 functions t : R n R and g j : R n R, j = 1,..., r, the primal problem P is as follows: In our case: g(s) = t(s) = min{t(x) : x X, g j (x) 0 for j = 1,..., r}. (6) w c h s i + λ i i=1 (1 L) MT BF L j W π ij (s) c tr ji + λ i P B (s) c em, (7) [ w λ i ] π ij (s) T ji + λ i P B (s) (MT T R + OS), (8) i=1 j W X = {s i 0, s i integer i W }. (9) 8

9 The Lagrangian dual D is defined as: max{q(γ) : γ 0}, (10) where q(γ) = inf{l(x, γ) : x X} and L(x, γ) = t(x) + γ g(x). Function q(γ) is called the Lagrangian relaxation of the primal problem P, L(x, γ) is the Lagrangian function and γ is the transpose of vector γ. The following properties hold [5]. The Lagrangian relaxation q(γ) is a concave function. Given any γ 0 and any feasible solution x for the primal problem, q(γ) t(x) holds. This property is called weak duality. Let x be a solution to the Lagrangian relaxation, i.e., such that q( γ) = L( x, γ). If x is feasible for the primal problem and γ g( x) = 0, then x is an optimal solution for the primal problem. In this case strong duality holds, i.e., q( γ) = t( x). The conditions γ g( x) = 0 are called complementary slackness conditions. Let x be an optimal solution for the primal problem and γ an optimal solution for the Lagrangian dual. The difference t(x ) q(γ ) (non-negative by weak duality) is called the duality gap between the primal problem and the Lagrangian dual. Let UB be an upper bound to t(x ). We denote UB q(γ ) a duality bound between the primal problem and the Lagrangian dual. It is clear than a duality bound is always larger than or equal to the duality gap. If t = q(γ ), then strong duality holds. Unfortunately, strong duality rarely occurs in integer programming. In this paper we use a method to strengthen the primal formulation in order to achieve strong duality. In subsection we deal with the exact computation of the Lagrangian relaxation q(γ) of problem P 0. Our approach is similar to that of Wong [19]. Then, in subsection 4.1.2, we use the sub-gradient method to find the maximum q(γ ) of the Lagrangian relaxation. A simple upper bound to the optimal primal solution is obtained heuristically in subsection Finally, in subsection 4.2, we deal with the computation of an optimal solution s to Problem P The Lagrangian relaxation For a given γ, the Lagrangian relaxation q P0 (γ) of problem P 0 is obtained by relaxing the operational availability constraint: 9

10 { w q P0 (γ) = min s {L(s, γ)} = min s i=1 ch s i + λ i j W π ij(s) c tr ji + λ i P B (s) c em + [ +γ (1 L) MT BF w L i=1 (λ i )]} j W π ij(s) T ji + λ i P B (s) (MT T R + OS) (11) Let ZP 0 be the optimum of problem P 0. Given an allocation s, let Z P0 (s) be the value of the objective function of problem P 0 for the given s, let OA(s) be the operational availability of the network under s, and MCMT (s) be the mean corrective maintenance time under s. From the properties of the Lagrangian relaxation shown in the previous section, it follows that [19]: Z P 0 q P0 (γ) for any γ 0 Z P 0 max γ 0 q P0 (γ) If for some γ 0 the optimal solution to problem min s {L(s, γ)} is s and s is a feasible allocation for problem P 0, then (Z P0 (s ) ZP 0 ) γ (MCMT (s (1 L) MT BF ) ( )) L If for some γ 0 the optimal solution s to problem min s {L(s, γ)} is feasible for P 0 and γ ( (1 L) MT BF L then s is an optimal solution to problem P 0. ) MCMT (s ) = 0, Let us now focus on the computation of q P0 (γ) = min s {L(s, γ)} for a given γ. Problem (11) can be rewritten as: { w q P0 (γ) = min s i=1 [c h s i + λ i j W π ij(s) (c tr ji γ T ji )+ ] } (12) +λ i P B (s) (c em γ (MT T R + OS)) + γ (1 L) MT BF L Note that the last term does not depend on s and can be omitted for the purpose of finding s. Moreover, the quantity [ ] w i=1 c h s i + λ i P B (s) (c em γ (MT T R + OS)) only depends on the overall stock level S = w i=1 s i (see Section 3.3). We can therefore define the functions f 1 (S) = w i=1 [ ] c h s i + λ i P B (s) (c em γ (MT T R + OS)) f 2 (s) = w λ i i=1 j W π ij (s) (c tr ji γ T ji ) 10

11 and rewrite the problem as: min s {f 1 (S) + f 2 (s)} (13) This problem can be solved at optimality by using a procedure by Wong [19]. The idea of the procedure is to introduce the quantities f 3 (S) = min s {f 2 (s) : j W s i = S} and to solve the problem min S {f 1 (S) + f 3 (S)} for increasing values of S, starting from S = The sub-gradient optimization procedure When problem (13) is solved for a given γ, we can easily compute q P0 (γ), a lower bound on Z P 0. In order to solve the Lagrangian dual (10) we use the sub-gradient optimization method. Note that, letting sˆγ = arg min s {f 1 (S) + f 2 (s)}, the function g(sˆγ ) defined by equation (8) provides a sub-gradient of function q(γ) for γ = ˆγ. The sub-gradient method generates a sequence of dual feasible points obtained from an initial value γ 0, for k = 0, as follows: γ k+1 = [γ k + o k g k ] + (14) Here, g k = g(s γ k) denotes the sub-gradient at point γ k, [ ] + denotes projection on the set {γ : γ 0, q P0 (γ) > }, and o k is a positive scalar stepsize computed as in [19]. o k = σ k q(γk ) Ẑ g(s γ k) 2 2 In this formula, Ẑ is the the best upper bound to problem P 0 so far. It is initially computed with the algorithm described in subsection and then updated during the procedure execution. The value σ k is a scalar chosen between 0 and 2. A sketch of the sub-gradient method is reported in algorithm Upper-bound computation A simple upper bound to Z P 0 is computed by allocating spare parts only to those warehouses i W with positive demand λ i > 0 and by avoiding concentration in just few warehouses. Simulation experiments [7] show that this allocation policy is effective in many practical contexts. Therefore its aim is that of allocating parts giving preference to warehouses with larger demand. 11

12 Algorithm 1 Sub-gradient Algorithm 1: k 0, γ 0 0, Ẑ solution of heuristics described in subsection : while sub-gradient direction NOT close to 0 and lower bound value not respecting any Lagrangian value constraint (refer to section 4.2) do 3: for all i W do 4: s i = 0, s i = 0 5: end for 6: min t(0), refer to equation 7; 7: go = true 8: while go == true do 9: S = S + 1, compute f(s), referring to 13; 10: if f(s) min then 11: go = false 12: else 13: for all possible stock allocations ŝ with ŝ ŝ w = S and respecting any objective level constraint (refer to section 4.2) do 14: calculate f(s) and g(ŝ) 15: if f(s) + g(ŝ) < min then 16: min = f(s) + g(ŝ), s i = ŝ i i = 1,..., w 17: end if 18: if ŝ feasible for problem P 0 5 and t(ŝ) < Ẑ, by using equation 7 then 19: Ẑ t(ŝ) 20: end if 21: end for 22: end if 23: end while 24: k = k + 1; 25: compute γ k and the new sub-gradient direction, refer to 14, by using s ; 26: end while 12

13 Our heuristic procedure greedily allocates one spare at a time and checks the operational availability of system. The warehouses i W with λ i > 0 are ordered for decreasing value of λ i, and a spare is allocated to each warehouse in this order until the contractual level of the Operational Availability OA is met. 4.2 Convergent Lagrangian and objective level cut method In order to achieve an optimal allocation s for problem P 0 we follow the convergent Lagrangian and objective level cut method described in [15] with slight modifications to adapt the method to our single echelon spare part optimization problem. The objective level cut is used to eliminate the duality gap and thus to guarantee the convergence of the Lagrangian method on a revised domain Objective level cut method for the spare optimization problem In this section, we describe the solution scheme of the convergent Lagrangian and objective level cut method for solving problem P 0. This method has been proposed and applied in [15] for separable non linear integer programming and polynomial 0-1 programming. In [15] an integer objective value is required for optimality, whereas the objective function of problem P 0 may not be integer. However, we show that the method can be adapted to deal with problem P 0. The key ideas consist of replacing exact lower and upper bound values with their respective integer rounded up values and adding Lagrangian level cuts. We can prove that the properties of the Lagrangian and objective level cut method still applies, and that the proposed algorithm, sketched in algorithm 2, finds an optimal solution to Problem P 0, or proves infeasibility, in at most UB 1 LB iterations (see theorem 3). UB 1 and LB 1 are respectively the upper and lower bound values computed by solving the conventional Lagrangian dual problem (10). Therefore, the efficiency of the algorithm depends on the solution algorithm chosen for solving the Lagrangian dual in the revised form, where objective and Lagrangian level cuts have been added. The algorithm starts solving the conventional Lagrangian dual problem. In case of a positive duality gap, a revised dual problem is iteratively formulated and solved until the duality gap reduces down to zero. Specifically, a lower objective level cut is added to problem P 0, thus reducing the duality gap (see theorems 1 and 2), and a lower Lagrangian cut is imposed to the Lagrangian dual problem (10), so that the Lagrangian dual value is increased with respect to the current lower bound and the duality gap is further reduced (see theorem 2). 13

14 Note that the optimal solution is always valid for both cuts. Let l and u be a lower and an upper bounds to problem (5), respectively. Then the revised problem is as follows: Revised problem P 0 (l, u): min t(s) = w i=1 [ch s i + λ i j W π ij(s) c tr ji + λ i P B (s) c em ] s.t. : w i=1 [λ i j W π ij(s) T ji + λ i P B (s) (MT T R + OS)] > (1 L) MT BF L s X(l, u) = {s : s i 0, s i integer, i W, l t(s) u } Clearly, P 0 (l, u) is equivalent to P 0 when l t u. The Lagrangian relaxation of P 0 (l, u) is: Revised problem q (l,u) P 0 (γ): min s {L(s, γ)} = min s { w i=1 ch s i + λ i j W π ij(s) c tr ji + λ i P B (s) c em + +γ [ ( w i=1 [λ i j W π ij(s) T ji + λ i P B (s) (MT T R + OS)])+ (1 L) MT BF + } L (15) s X(l, u) {s : s i 0, s i integer, i W, L(s, γ) > l } The Lagrangian dual problem of P 0 (l, u) is then given as: Revised Lagrangian dual problem D(l, u): max q (l,u) P 0 (γ) s.t. : γ 0 (16) (17) In case of duality gap the algorithm starts solving the revised Lagrangian dual problems by using as initial lower and upper bound values the ones computed by solving the conventional Lagrangian dual problem (10). Specifically, u = q(γ) and l = t(s), computed according to equations (10) and (7), the latter firstly generated heuristically (as described in subsection 4.1.3) and updated eventually during the subgradient search. Then, the method updates these lower and upper bound values iteratively, by adding new objective and Lagrangian cuts, computed by solving problem (17). Given a Lagrange multiplier ˆγ, and since the dual function q (l,u) P 0 (γ) described in (17) is a concave function, its maximum can be computed through sub-gradient optimization, as described in subsection A sketch of the overall 14

15 Algorithm 2 Convergent Lagrangian and objective level cuts algorithm 1: LB 0 and UB 0 Ẑ (Ẑ feasible solution for P 0, found by applying heuristics described in subsection 4.1.3) 2: k = 0, go = true 3: while go do 4: k = k + 1, solve D( LB k, UB k ), for computing the best γ k ; 5: LB q ( LB k, UB k ) (γ k ) and UB k Ẑ (Ẑ best feasible solution for P 0 ( LB k, UB k ), found during the dual search) 6: if LB k = UB k then 7: go = false 8: else 9: go = true 10: end if 11: if UB k < LB k then 12: go = f alse, problem infeasible 13: end if 14: end while method is provided by Algorithm 2. It can be proved [15] that at least one infeasible solution will be cut when adding a new objective cut. Theorem 1 The following properties hold: 1. Let γ (l, u) denote the optimal solution to the Lagrangian dual (10). The optimal dual value q P0 (γ, l, u) is a non decreasing function of l 2. If l t u then q P0 (γ ) q P0 (γ, l, u) t. Moreover let σ = max{t(s) t(s) < t, x X(l, u)}. If σ < l t, then γ = 0 and q P0 (γ, l, u) = t 3. For l < t, we have q P0 (γ, l, u) l Proof. The proof is similar to that provided in [14] for an analogous theorem. The only difference is the introduction of rounded up values for the upper and lower bounds, which does not affect the proof. Moreover we have: Theorem 2 Let Q P0 (γ ) be the set of feasible solutions of q P0 (γ ). If q P0 (γ, l, u) < t, then min{f(s) s Q P0 (γ ), g(s) > 0} q P0 (γ, l, u) holds. 15

16 Proof. The proof is similar to that provided in [14] for an analogous theorem. The only difference is the introduction of rounded up values for the upper and lower bounds, which does not affect the proof. Theorem 1 implies that the revised dual problem provides a value which is greater or equal to the current lower bound. Theorem 2 guarantees that at least one infeasible solution is cut by an objective cut higher than the current lower and upper bounds l and u. The introduction of rounded up values for lower and upper bounds allows us to prove the following theorem: Theorem 3 Algorithm 2 either finds an optimal solution of problem 5 or reports an infeasibility of problem 5 in at most UB 1 LB iterations. Proof. From Algorithm 2, it follows that LB k t holds. If the algorithm stops at step 12 problem 5 is infeasible. Optimality holds when Algorithm 2 stops at step 7. If the algorithm does not stop at iteration k, then LB k+1 LB k + 1 holds at each step. Notice that for any k, LB k t UB k + 1 holds. Therefore, in at most UB 1 LB 1 iterations UB k = LB k must hold. If the algorithm does not stop before UB 1 LB iterations, then it must stop at UB 1 LB th iteration either with an infeasibility or reaching the optimal solution. 5. Case study from the corrective airport maintenance context In this section we evaluate the effectiveness of our method for a practical problem arising in the airport maintenance context. We analyze 30 practical instancesprovided by an Italian logistics company supporting the activity of the 38 civil airports spread over the Italian territory. The company handles 17 warehouses and manages the overall processes of purchasing, holding, ensuring that the overall reliability of safety equipments is always within contractual limits. The aim of the company is therefore to grant the prescribed quality of service at minimum cost. While the company currently follows a two echelon policy for spare part management, in this paper we evaluate the potential of the single echelon policy, which is generally acknowledged to achieve better performance in this context. 16

17 The instances differ in the mean demand at each warehouse and in the number of warehouses with positive demand. The replenishment time of an item has been set equal to three months for all items and warehouses. Table 1 reports the main data for our instances. Item Cost Replenishment time MTBF number airports number items months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h months h Table 1: Characteristic data for 30 items A proven optimum has been computed for each instance by using a complete enumeration technique. Starting from S = 1, and then increasing S by one at each step, the costs associated to all possible stock allocations s with s s w = S are evaluated until a feasible stock allocation is found for some S. We can conclude at this point that no better solutions can be obtained for larger values of S, since purchasing and holding costs are increasing 17

18 with S and dominate the transshipment costs. For each instance, we compute the same stock allocation either by using our algorithm or by using the enumerative technique. In all cases, the algorithm finds a solution with the same proven minimum cost of the enumeration technique. Each optimal solution has been found within one minute of computation time, whereas the computation time required by the enumeration technique is sometimes larger than 30 minutes. 6. Conclusions In this paper we propose and evaluate a solution methodology for optimizing inventory stock allocation of repairable spare parts in a single echelon, w-locations system, where lateral and emergency shipments occur in response to stockouts. We model our problem as a nonlinear integer program and apply a primal based, dual based algorithm for allocating the spare parts optimally. We evaluate this technique by using practical data from airport corrective maintenance context, by determining the stocking policy for items supporting the daily activities of 38 Italian airports. The technique finds optimal stock levels for all test instance for which the proven optimum can be found via total enumeration, thus demonstrating its validity. References [1] Airports Council International (ACI). Worldwide and Regional Forecasts, Airport Traffic , (2005). [2] Alfredsson, P. and J. Verrijdt. Modeling emergency supply flexibility in a two echelon inventory system. Management Science 45, (1999) [3] Axsater,S.. Modelling emergency lateral transshipments in inventory systems. Management Science 6, (1990) [4] Axsater,S.. Inventory control. Springer Verlang (2006). [5] Bertsekas, D.P.. Nonlinear programming. Athena Scientific second edition (2003). [6] Cesaro, A., Pacciarelli, D.. Performance assessment for single echelon airport spare part management. Technical Report University Roma Tre (2009). 18

19 [7] Cesaro, A., Pacciarelli, D.. Evaluation of peaked lateral transshipment in inventory system subject to a service constraint. MSOM 2007 conference (June 18 19,2007). [8] Dada, M.. A two echelon inventory system with priority shipments. Management Science 38, (1992) [9] Grahovac, J., A. Chakravarty. Sharing and Lateral Transshipment of Inventory in a Supply Chain with Expensive Low-Demand Items. Management Science 47, (2001) [10] Huiskonen, J.. Maintenance spare parts logistics: special characteristics and strategic choices. Int. J. Production Economics 71, (2001) [11] Kennedy, W.J., J. Wayne Patterson, and L. D. Fredendall. An overview of recent literature on spare parts inventories. Int. J. Production Economics 76, (2002) [12] Kukreja, A., C.P. Schmidt, D.M. Miller. Stocking decisions for low-usage items in a multilocation inventory system Management Science 47, (2001) [13] Kukreja, A., C.P. Schmidt A model for lumpy demand parts in a multi-location inventory system with transshipments. Computers and Operations Research 32, (2005) [14] Li, D., X.L. Sun. Towards strong duality in integer programming. J. Glob. Optim 35, (2006) [15] Li, D., J. Wang, X.L. Sun. Computing exact solution to nonlinear integer programming: Convergent Lagrangian and objective level cut method. J. Glob. Optim 39, (2007) [16] Li, D., X.L. Sun. Nonlinear integer programming. Springer s INTERNATIONAL SE- RIES (2006) [17] Sherbrooke, C.C.. Optimal inventory modeling of systems: multi-echelon techniques. Wiley (2004). [18] Taragas, G., D. Vlachos. Effectiveness of stock transshipment under various demand distributions and nonnegligible transshipment times. Production And operations Management 11, (2002)

20 [19] Wong, H.,D. Catrysse,D. Van Houdeusden, Inventory pooling of repairable spare parts with non-zero lateral transshipment time and delayed lateral transshipments. European Journal of Operational Research 165 (2005) [20] Wong, H.,D. Catrysse,D. Van Houdeusden. Stocking decisions for repairable spare parts pooling in a multi-hub system. Int. J. Production Economics 93 (2005) [21] Wong, H.,G.J. Van Houtum, D. Catrysse,D. Van Houdeusden. Simple, efficient heuristics for multi-item multi-location spare parts systems with lateral transshipments and waiting time constraints. Journal of the Operational Research Society 56 (2005),